Twisted modules from coherent gluing

Written by GPT-6.1 Sol (OpenAI), reasoning effort Ultra, September 2026. Self-checked; no independent review. Public domain (CC0).

An ordinary module sheaf glues by maps whose composite on a triple overlap is the direct transition map. A twisted module allows a controlled modification of that law. The modification itself must be coherent on quadruple overlaps. The result is a category of modules that is ordinary locally, even when no global trivialization exists.

The prerequisites are Representing objects from local data and Local module categories and Morita equivalence. We also assume the definitions of a site, a sheaf and a covering. We recall the part of stack descent used in the construction. Basic references are [D'Agnolo–Polesello], [Stacks], and [Schapira, Sheaves].

1. From a cover to a multiplication law

Work in the topos \(\mathcal E\) of sheaves on a small site. Let \(R\) be a sheaf of commutative unital rings. All products below are products in \(\mathcal E\); thus the construction also works when the site has no object representing a particular product. For a sheaf \(V\), module sheaves over \(V\) mean modules in the slice topos \(\mathcal E/V\), over the restriction of \(R\).

Let \(\pi:U\to1\) be an epimorphism. It is a cover of the terminal sheaf. A covering family of site objects gives one by taking the coproduct of their associated sheaves. Epimorphisms of sheaves have sections locally: a section of their target lifts after a covering. Consequently the pulled-back cover \(U\times V\to V\) has a section locally on \(V\).

A line in this lesson is an invertible \(R\)-module: a module \(L\) with a tensor inverse \(L^*\). We do not assume every such module is locally free on the given site. For a locally ringed space, it is a line bundle in the familiar sense.

Choose an invertible module \(L\) over \(U\times U\), together with an isomorphism over \(U^3\)

\[ \mu:L_{12}\otimes_R L_{23}\longrightarrow L_{13}. \tag{1.1} \]

Subscripts indicate pullback along the relevant projection. On \(U^4\), require

\[ \mu_{134}(\mu_{123}\otimes\mathrm{id}) =\mu_{124}(\mathrm{id}\otimes\mu_{234}). \tag{1.2} \]

Both sides take \(L_{12}\otimes L_{23}\otimes L_{34}\) to \(L_{14}\). Parentheses are identified by the canonical tensor associator.

There is a canonical unit \(R_U\simeq\Delta^*L\), where \(\Delta:U\to U^2\) is the diagonal. Here is why. Restrict (1.1) to the triple diagonal to obtain an associative isomorphism \(m:D\otimes D\to D\) on the invertible module \(D=\Delta^*L\). Tensoring with \(D\) is an equivalence. There is therefore a unique map \(e:R\to D\) with \(m(e\otimes\mathrm{id})=\mathrm{id}_D\). Associativity implies \(m(\mathrm{id}\otimes e)=\mathrm{id}_D\): compare the two maps after tensoring on the left with \(D\), use the first unit identity, then cancel that equivalence. Since \(m\) and tensoring with \(D\) are isomorphisms, \(e\) is an isomorphism. Restricting (1.2) to repeated coordinates gives the same unit identities for \(L(x,y)\). We will use this canonical unit without additional choices.

For a topological open cover \((U_i)\), these data become lines \(L_{ij}\) on \(U_i\cap U_j\) and associative isomorphisms \(L_{ij}\otimes L_{jk}\simeq L_{ik}\). Invertibility and (1.1) identify \(L_{ji}\) as a tensor inverse of \(L_{ij}\).

2. The twisted category

For any \(V\in\mathcal E\), define \(\mathsf T(V)\) as follows. An object is an \(R\)-module \(M\) over \(U\times V\) and an isomorphism over \(U^2\times V\)

\[ \theta:L_{12}\otimes M_2\longrightarrow M_1 \tag{2.1} \]

such that over \(U^3\times V\)

\[ \theta_{12}(\mathrm{id}\otimes\theta_{23}) =\theta_{13}(\mu_{123}\otimes\mathrm{id}). \tag{2.2} \]

A morphism \((M,\theta)\to(N,\psi)\) is a module map \(a:M\to N\) with

\[ a_1\theta=\psi(\mathrm{id}_L\otimes a_2). \tag{2.3} \]

Pullback in \(V\) defines the restriction functors. They carry the canonical coherence isomorphisms of pullback, so this is a prestack. We use stack for a prestack in categories whose morphisms are sheaves and whose descent data for objects are effective. Morphisms inside a category need not be invertible. The comparison maps used to glue objects are invertible.

The diagonal restriction of \(\theta\) is the identity after identifying \(\Delta^*L\) with \(R\). In fact (2.2) on the diagonal says that its invertible endomorphism \(h\) satisfies \(h^2=h\). Multiplying by \(h^{-1}\) gives \(h=\mathrm{id}\).

Theorem 2.1. The prestack \(\mathsf T\) is an \(R\)-linear stack and is locally equivalent to the stack of \(R\)-modules.

Proof of descent. Let \((V_a\to V)\) be a cover, and suppose objects of \(\mathsf T(V_a)\) have isomorphisms on pairwise overlaps satisfying the usual cocycle law. Their underlying module sheaves glue over \(U\times V\). One explicit description of the glued sheaf is compatible local sections after pulling back the cover to each test object; restrictions and addition act coordinatewise. The sheaf axiom gives uniqueness, and the local module actions glue because their compatibility is one of the local equalities. The maps \(\theta_a\) glue to \(\theta\), since (2.3) says they agree under the gluing maps. Its inverse glues too. Equation (2.2) is an equality of sheaf maps and can be checked after the cover, where it holds. This proves effectiveness for objects. The same compatible-section description glues morphisms uniquely and preserves (2.3). Hence the morphism presheaves are sheaves. Linearity is inherited from module morphisms. \(\square\)

3. Trivialize using a section of the cover

We finish the proof of Theorem 2.1 by constructing the local equivalence explicitly. Suppose the cover over \(V\) has a section \(s:V\to U\). For an ordinary module \(N\) over \(V\), put

\[ \Phi_s(N)_{(x,v)}=L(x,s(v))\otimes_R N_v. \tag{3.1} \]

This notation abbreviates pullback of \(L\) by \((x,v)\mapsto(x,s(v))\), tensor pullback of \(N\) by \(U\times V\to V\). It does not require points of the topos. The transition isomorphism is

\[ L(x,y)\otimes L(y,s(v))\otimes N_v \xrightarrow{\mu\otimes\mathrm{id}} L(x,s(v))\otimes N_v. \]

Equation (1.2) is exactly (2.2) for this object. Define \(E_s(M,\theta)=s^*M\), pulling back along \(v\mapsto(s(v),v)\). The unit of \(L\) gives \(E_s\Phi_s(N)\simeq N\). Conversely, restricting \(\theta\) to \((x,s(v),v)\) gives

\[ \Phi_s E_s(M,\theta)\simeq(M,\theta). \tag{3.2} \]

It is a morphism in \(\mathsf T\) by (2.2). All these maps are natural and commute with further restriction. Thus \(E_s,\Phi_s\) are inverse equivalences of stacks over \(V\). Sections \(s\) exist locally, proving local equivalence everywhere. \(\square\)

Two choices \(s,t\) differ by an ordinary invertible module:

\[ E_t\Phi_s(N)=L(t,s)\otimes_R N. \tag{3.3} \]

Thus a change of local description is a Morita equivalence. It need not be the identity functor.

4. Every locally ordinary module stack has this form

Theorem 4.1. Let \(\mathsf S\) be an \(R\)-linear stack locally equivalent to \(\mathsf{Mod}(R)\). After choosing a cover and local equivalences, it is equivalent to a stack \(\mathsf T\) constructed above.

Proof. Choose a cover \(U\to1\) and an equivalence \(F:\mathsf S|_U\to\mathsf{Mod}(R)|_U\) with an adjoint inverse. On \(U^2\), the change from the second trivialization to the first is \(F_1F_2^{-1}\). By Theorem 4.1 of Local module categories and Morita equivalence, it is tensoring with

\[ L_{12}=(F_1F_2^{-1})(R). \]

The two scalar actions agree because the functor is \(R\)-linear. Its Morita inverse is consequently an ordinary tensor inverse of this \(R\)-module.

On \(U^3\), cancel the middle inverse equivalence in \(F_1F_2^{-1}F_2F_3^{-1}\). Evaluating the resulting natural isomorphism to \(F_1F_3^{-1}\) at \(R\) gives \(\mu:L_{12}\otimes L_{23}\simeq L_{13}\). On \(U^4\), cancel the two middle pairs in either order. The two counits act in distinct positions, so naturality makes the two cancellations equal. The associativity constraints for composition give (1.2); the adjunction triangle identities give the diagonal unit.

An object of \(\mathsf S(V)\), restricted to \(U\times V\) and sent through \(F\), gives \(M\). Its canonical descent isomorphism, expressed in the two trivializations, gives \(\theta\). The cocycle law for that descent isomorphism gives (2.2) by the construction of \(\mu\). A morphism becomes (2.3).

Conversely, apply \(F^{-1}\) locally to \(M\). Equation (2.2), with the same cancellation isomorphisms, is precisely the ordinary cocycle law for these local objects of \(\mathsf S\). They glue because \(\mathsf S\) is a stack. The sheaf condition for morphisms gives full faithfulness. These procedures are inverse up to the chosen equivalences and their units and counits, and are compatible with restriction. \(\square\)

There is also a canonical identification

\[ \mathcal Aut(\mathrm{id}_{\mathsf S})\simeq R^\times. \tag{4.1} \]

Locally on an ordinary module category, a natural endomorphism of the identity is multiplication by a section of \(R\). To see this, evaluate it at \(R\); naturality for \(R\to M\), \(1\mapsto m\), determines it on every local section of every module. It is invertible exactly for a unit. Tensoring with an invertible module preserves that scalar action, so the local identifications agree and glue. No chosen trivialization remains in (4.1).

Theorem 4.2. Every \(R\)-linear autoequivalence \(H\) of \(\mathsf S\) is tensoring with an invertible \(R\)-module. That module is uniquely determined up to isomorphism.

Proof. Let \(P=\mathcal Nat(\mathrm{id}_{\mathsf S},H)\), the sheaf of natural transformations of these stack functors. It is an \(R\)-module sheaf. Its components glue because morphisms in \(\mathsf S\) are sheaves, and naturality is a local equality of maps.

In an ordinary module description, the sheaf Morita theorem writes \(H\) as \(P_0\otimes_R-\) for an invertible module \(P_0\). Evaluation at \(R\) identifies \(\mathcal Nat(\mathrm{id},P_0\otimes_R-)\) with \(P_0\): a section \(p\) acts by \(m\mapsto p\otimes m\), and naturality for every local map \(R\to M\) determines all its components. Thus \(P\) is locally invertible. The local evaluation maps \(P\otimes_R P^*\to R\), where \(P^*=\mathcal Hom_R(P,R)\), are isomorphisms; hence \(P\) is globally invertible.

Evaluation of transformations, followed by the tensor universal property of Corollary 3.1 in Representing objects from local data, gives \(P\otimes_R K\to H(K)\). In each ordinary module description it is the tensor isomorphism just obtained, so it is an isomorphism everywhere. Conversely, if \(H\simeq Q\otimes_R-\), the same local calculation identifies its sheaf of natural transformations from the identity with \(Q\). This proves uniqueness. \(\square\)

5. Scalar cocycles and their changes

On a topological open cover, suppose each \(L_{ij}\) is given a basis. Then \(\mu\) is multiplication by a unit \(c_{ijk}\in R^\times(U_i\cap U_j\cap U_k)\). Equation (1.2) becomes

\[ c_{ijk}c_{ikl}=c_{jkl}c_{ijl}. \tag{5.1} \]

The twisted module maps \(t_{ij}:M_j\to M_i\) satisfy

\[ t_{ij}t_{jk}=c_{ijk}t_{ik}. \tag{5.2} \]

All identities involving a repeated index are normalized by the unit. This is a concrete example of the preceding construction, rather than an assumption that all lines on all sites have bases.

If units \(b_{ij}\) change the transition maps to \(t'_{ij}=b_{ij}t_{ij}\), then

\[ c'_{ijk}=b_{ij}b_{jk}b_{ik}^{-1}c_{ijk}. \tag{5.3} \]

Thus a coboundary change gives an equivalent twisted category. For an example, choose arbitrary units \(a_i\) on \(U_i\) and put \(b_{ij}=a_i a_j^{-1}\). The resulting coboundary is one, so these changes preserve \(c\). They act by changing local frames, without changing the coherence law.

Here is a numerical coherence check. Cover a point by four labeled copies of itself and take \(R=\mathbb Q\). Set \[ b_{01}=2,\quad b_{12}=3,\quad b_{02}=5,\quad b_{23}=7,\quad b_{03}=11,\quad b_{13}=13, \] with \(b_{ii}=1\) and \(b_{ji}=b_{ij}^{-1}\). Define \(c_{ijk}=b_{ij}b_{jk}b_{ik}^{-1}\). The four factors on the overlap labeled \((0,1,2,3)\) are \[ c_{012}=\frac65,\quad c_{023}=\frac{35}{11},\quad c_{123}=\frac{21}{13},\quad c_{013}=\frac{26}{11}. \] Both products in (5.1) are \(42/11\). Taking \(M_i=\mathbb Q\) and \(t_{ij}=b_{ij}\) gives a twisted module with these nonidentity factors. The twist is nevertheless a coboundary, as its definition shows; this example demonstrates coherence, without claiming a nontrivial cohomology class.

The Čech description depends on the cover and the existence of the chosen bases. A classification by sheaf cohomology requires a further comparison with gerbes and refinements. Equations (5.1)–(5.3) alone do not establish that comparison.

6. Exercises with solutions

Exercise 6.1 (foundation). If \(L\) is the trivial line and \(\mu\) its ordinary multiplication, identify \(\mathsf T\).

Solution. Equation (2.1) becomes an ordinary descent isomorphism \(M_2\to M_1\) over the cover \(U\times V\to V\). Equation (2.2) is its cocycle law. Effective descent for module sheaves glues \(M\) to a unique module over \(V\), up to its canonical isomorphism; morphisms glue uniquely as well. Hence \(\mathsf T\simeq\mathsf{Mod}(R)\).

Exercise 6.2 (calculation). In the scalar setting, compose two changes \(b_{ij}\) and \(d_{ij}\). Check the resulting coherence factor and the inverse change.

Solution. The new maps are \(d_{ij}b_{ij}t_{ij}\). Their factor is \[ (d_{ij}d_{jk}d_{ik}^{-1})(b_{ij}b_{jk}b_{ik}^{-1})c_{ijk}. \] Commutativity allows the factors to regroup as the coboundary of \((d_{ij}b_{ij})\) times \(c\). Replacing \(b\) by \(b^{-1}\) gives the inverse. This also verifies that the equivalence on morphisms leaves (2.3) intact.

Exercise 6.3 (obstruction). Suppose a scalar-twisted module is locally free of constant positive rank \(r\). What does taking determinants in (5.2) imply? What is special about rank one?

Solution. Write \(d_{ij}=\det t_{ij}\) after choosing local bases. Then \(d_{ij}d_{jk}=c_{ijk}^{r}d_{ik}\). Thus \(c^r\) is the coboundary of the unit cochain \(d\). In particular the Čech class of \(c\) has order dividing \(r\). For rank one, \(c\) itself is a coboundary, so rescaling transitions turns this particular twist into ordinary descent. This is a necessary condition for an object of the stated rank; no existence of such an object follows from a torsion condition alone.

Exercise 6.4 (synthesis). For sections \(s,t,u\) of the covering over \(V\), compare \(E_u\Phi_t E_t\Phi_s\) with \(E_u\Phi_s\). Identify the map that makes this comparison coherent for four sections.

Solution. Formula (3.3) gives the first functor as tensoring with \(L(u,t)\otimes L(t,s)\), and the second as tensoring with \(L(u,s)\). Their comparison is \(\mu(u,t,s)\). With four sections, the two comparisons multiply three line factors in opposite parenthesizations. Equation (1.2) makes these maps equal. Hence the local equivalences carry the full composition coherence, in addition to being pairwise equivalences.

References